How to smoothly combine objects at a right angle?
I personally think the imperfections in your joined planks look good.

However for a precise approach consider the following. The Knife Tool has the option to cut in steps of 45°.


Go into edit mode. &#8633; Tab
Use the Knife Tool K to cut a 45 degree angle by pressing C once to activate snapping duration the operation.
Remove the vertices which are not needed. X
Snap the cursor to the outer corner where the planks join. &#8679; ShiftS > Cursor to selected
Select all A and duplicate the geometry &#8679; ShiftD. Let it rest in the same place.  RMB
Activate Cursor as Pivot NumpadPeriod, then rotate the duplicated geometry 90 degree around the cursor pivot. R90&#9166; Enter.
Mirror it in Y. SY-1 or &#9096; CtrlMY
Remove the double vertices. W>R